Volatility: Classification Systems and Dispersion Indices

Volatility in the context of analytical systems is defined as a measure of the dispersion of random variable values relative to their mathematical expectation. Formally, volatility is related to the variance D[X] = E[(X − μ)²] = E[X²] − (E[X])², where μ = E[X] is the mathematical expectation. Unlike financial markets, where volatility is measured as the standard deviation of logarithmic returns, in systems of discrete outcomes a direct calculation of the variance from the payout table is applied. Volatility is a key parameter determining the nature of short-term deviations from the expected value.

The classification of volatility involves dividing systems into categories: low (σ/μ < 3), medium (3 ≤ σ/μ < 8), high (8 ≤ σ/μ < 15), and extreme (σ/μ ≥ 15), where σ/μ is the coefficient of variation. The Volatility Index (VI) is calculated as a normalized indicator that takes into account not only the variance, but also the skewness and excess kurtosis of the distribution. The index formula is: VI = w₁·CV + w₂·|γ₁| + w₃·(γ₂ − 3), where CV is the coefficient of variation, γ₁ is the skewness coefficient, γ₂ is the kurtosis, and wᵢ are normalizing weight coefficients. Such a multidimensional classification allows for a more accurate characterization of the system’s risk profile.

The distinction between variance and volatility is of fundamental importance for the correct interpretation of results. Variance is a strictly defined statistical measure of the second central moment of the distribution, measured in the square of the units of the initial variable. Volatility, on the contrary, is a broader concept that includes not only variance but also the characteristics of the distribution tails. For heavy-tailed distributions, typical of systems with rare large multipliers, the standard variance may be uninformative, and alternative metrics are applied: MAD (median absolute deviation) or CVaR (conditional value at risk).

The calculation of the dispersion index for a specific system is carried out on the basis of a complete payout table, taking into account the probabilities of each outcome. The algorithm includes the following steps: (1) calculating the mathematical expectation μ = Σpᵢmᵢ; (2) calculating the second moment E[X²] = Σpᵢmᵢ²; (3) determining the variance D = E[X²] − μ²; (4) calculating the standard deviation σ = √D; (5) calculating the coefficient of variation CV = σ/μ. To verify the theoretical calculation, a comparison is made with the empirical variance obtained by the Monte Carlo method on a sample of at least 10⁷ iterations.